By Andrei Y. Khrennikov

N atur non facit saltus? This publication is dedicated to the basic challenge which arises contin uously within the technique of the human research of truth: the position of a mathematical equipment in an outline of fact. We pay our major consciousness to the position of quantity platforms that are used, or can be utilized, during this approach. we will express that the image of truth in keeping with the normal (since the works of Galileo and Newton) equipment of genuine research isn't the certain real way of featuring fact in a human mind. There exist different photographs of fact the place different num ber fields are used as simple components of a mathematical description. during this booklet we attempt to construct a p-adic photo of fact in keeping with the fields of p-adic numbers Qp and corresponding research (a specific case of so known as non-Archimedean analysis). although, this ebook mustn't ever be regarded as just a e-book on p-adic research and its purposes. We research a way more prolonged variety of difficulties. Our philosophical and actual rules could be learned in different mathematical frameworks which aren't obliged to be according to p-adic research. we will exhibit that many difficulties of the outline of truth by way of actual numbers are triggered by means of limitless purposes of the so referred to as Archimedean axiom.

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In the same way we cannot identify geometry with the model described by the Euclidean axiomatic. Further, we shall consider non-Kolmogorov probabilistic models. These models provide a new point of view of the unit and zero probabilities. These notions depend on a probabilistic model. An event which has unit probability from the point of view of one observer (one probabilistic model) may have zero probability from the point of view of another observer (another probabilistic model). Therefore the EPR definition depends on the Kolmogorov probabilistic model.

We could never measure the original segment L with the aid of our unit of measurement I. , a violation of the Archimedean axiom (AJ. However, these heuristic considerations have not been realized in a rigorous form because we cannot do this in the framework of M R 8. ) we have the image of space and time as infinitely deep and infinitely divisible structures. This idea is very old and attractive. It has worked fruitfully for centuries. However, this idea is the root of many paradoxes in applications of the MR model.

Denote by Z(F) the ring generated in F by its unity element. , n·1 = 1 + ... + 1 #- 0 for any n = 1,2, .... ), = 9By a ring we always mean a commutative ring with identity 1. 20 Chapter 1 then Z(F) is isomorphic to the ring of integers Z. Therefore in this case we can consider Z as a subring of F. In what follows we consider only normed rings F which have zero characteristic. To illustrate how we can work with the strong triangle inequality we present two simple results. 1. Let I . IF be a non-Archimedean norm.